Tp tuning: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-21 22:05:04 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-21 23:43:46 UTC</tt>.

: The original revision id was <tt>~~354247026~~</tt>.

: The original revision id was <tt>354253296</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings of S since T is the Lp tuning. By the [[http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem|Hahn–Banach theorem]], Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a [[http://www.math.unl.edu/~s-bbockel1/928/node25.html|corollary of Hahn-Banach]], the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.

The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, Ƹ+J*, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.

The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.

Note that while the Hahn-Banach theorem is usually proven using the axiom of choice and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.

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Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings of S since T is the Lp tuning. By the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem" rel="nofollow">Hahn–Banach theorem</a>, Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a <a class="wiki_link_ext" href="http://www.math.unl.edu/~s-bbockel1/928/node25.html" rel="nofollow">corollary of Hahn-Banach</a>, the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.

The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, Ƹ+J*, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.

The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.

Note that while the Hahn-Banach theorem is usually proven using the axiom of choice and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-21 22:05:04 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-21 23:43:46 UTC</tt>.<br>
-: The original revision id was <tt>354247026</tt>.<br>
+: The original revision id was <tt>354253296</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 25: / Line 25: @@
 Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings of S since T is the Lp tuning. By the [[http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem|Hahn–Banach theorem]], Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a [[http://www.math.unl.edu/~s-bbockel1/928/node25.html|corollary of Hahn-Banach]], the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.
-The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, Ƹ+J*, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.
+The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.
 Note that while the Hahn-Banach theorem is usually proven using the axiom of choice and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.
@@ Line 54: / Line 54: @@
   Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings of S since T is the Lp tuning. By the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem" rel="nofollow"&gt;Hahn–Banach theorem&lt;/a&gt;, Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a &lt;a class="wiki_link_ext" href="http://www.math.unl.edu/~s-bbockel1/928/node25.html" rel="nofollow"&gt;corollary of Hahn-Banach&lt;/a&gt;, the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.&lt;br /&gt;
 &lt;br /&gt;
-The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, Ƹ+J*, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.&lt;br /&gt;
+The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.&lt;br /&gt;
 &lt;br /&gt;
 Note that while the Hahn-Banach theorem is usually proven using the axiom of choice and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.&lt;br /&gt;